Theorem tendocnv 36310

Description: Converse of a trace-preserving endomorphism value. (Contributed by NM, 7-Apr-2014.)

Hypotheses

Ref	Expression
tendosp.h	⊢ 𝐻 = (LHyp‘𝐾)
tendosp.t	⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)
tendosp.e	⊢ 𝐸 = ((TEndo‘𝐾)‘𝑊)

Assertion

Ref	Expression
tendocnv	⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑆 ∈ 𝐸 ∧ 𝐹 ∈ 𝑇) → ◡(𝑆‘𝐹) = (𝑆‘◡𝐹))

Proof of Theorem tendocnv

Step	Hyp	Ref	Expression
1		simp1 1061	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑆 ∈ 𝐸 ∧ 𝐹 ∈ 𝑇) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
2		tendosp.h	. . . . . 6 ⊢ 𝐻 = (LHyp‘𝐾)
3		tendosp.t	. . . . . 6 ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)
4		tendosp.e	. . . . . 6 ⊢ 𝐸 = ((TEndo‘𝐾)‘𝑊)
5	2, 3, 4	tendocl 36055	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑆 ∈ 𝐸 ∧ 𝐹 ∈ 𝑇) → (𝑆‘𝐹) ∈ 𝑇)
6		eqid 2622	. . . . . 6 ⊢ (Base‘𝐾) = (Base‘𝐾)
7	6, 2, 3	ltrn1o 35410	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑆‘𝐹) ∈ 𝑇) → (𝑆‘𝐹):(Base‘𝐾)–1-1-onto→(Base‘𝐾))
8	1, 5, 7	syl2anc 693	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑆 ∈ 𝐸 ∧ 𝐹 ∈ 𝑇) → (𝑆‘𝐹):(Base‘𝐾)–1-1-onto→(Base‘𝐾))
9		f1ococnv1 6165	. . . 4 ⊢ ((𝑆‘𝐹):(Base‘𝐾)–1-1-onto→(Base‘𝐾) → (◡(𝑆‘𝐹) ∘ (𝑆‘𝐹)) = ( I ↾ (Base‘𝐾)))
10	8, 9	syl 17	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑆 ∈ 𝐸 ∧ 𝐹 ∈ 𝑇) → (◡(𝑆‘𝐹) ∘ (𝑆‘𝐹)) = ( I ↾ (Base‘𝐾)))
11	10	coeq1d 5283	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑆 ∈ 𝐸 ∧ 𝐹 ∈ 𝑇) → ((◡(𝑆‘𝐹) ∘ (𝑆‘𝐹)) ∘ ◡(𝑆‘𝐹)) = (( I ↾ (Base‘𝐾)) ∘ ◡(𝑆‘𝐹)))
12		simp2 1062	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑆 ∈ 𝐸 ∧ 𝐹 ∈ 𝑇) → 𝑆 ∈ 𝐸)
13	6, 2, 4	tendoid 36061	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑆 ∈ 𝐸) → (𝑆‘( I ↾ (Base‘𝐾))) = ( I ↾ (Base‘𝐾)))
14	1, 12, 13	syl2anc 693	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑆 ∈ 𝐸 ∧ 𝐹 ∈ 𝑇) → (𝑆‘( I ↾ (Base‘𝐾))) = ( I ↾ (Base‘𝐾)))
15	6, 2, 3	ltrn1o 35410	. . . . . . . . . 10 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) → 𝐹:(Base‘𝐾)–1-1-onto→(Base‘𝐾))
16	15	3adant2 1080	. . . . . . . . 9 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑆 ∈ 𝐸 ∧ 𝐹 ∈ 𝑇) → 𝐹:(Base‘𝐾)–1-1-onto→(Base‘𝐾))
17		f1ococnv2 6163	. . . . . . . . 9 ⊢ (𝐹:(Base‘𝐾)–1-1-onto→(Base‘𝐾) → (𝐹 ∘ ◡𝐹) = ( I ↾ (Base‘𝐾)))
18	16, 17	syl 17	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑆 ∈ 𝐸 ∧ 𝐹 ∈ 𝑇) → (𝐹 ∘ ◡𝐹) = ( I ↾ (Base‘𝐾)))
19	18	fveq2d 6195	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑆 ∈ 𝐸 ∧ 𝐹 ∈ 𝑇) → (𝑆‘(𝐹 ∘ ◡𝐹)) = (𝑆‘( I ↾ (Base‘𝐾))))
20		f1ococnv2 6163	. . . . . . . 8 ⊢ ((𝑆‘𝐹):(Base‘𝐾)–1-1-onto→(Base‘𝐾) → ((𝑆‘𝐹) ∘ ◡(𝑆‘𝐹)) = ( I ↾ (Base‘𝐾)))
21	8, 20	syl 17	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑆 ∈ 𝐸 ∧ 𝐹 ∈ 𝑇) → ((𝑆‘𝐹) ∘ ◡(𝑆‘𝐹)) = ( I ↾ (Base‘𝐾)))
22	14, 19, 21	3eqtr4rd 2667	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑆 ∈ 𝐸 ∧ 𝐹 ∈ 𝑇) → ((𝑆‘𝐹) ∘ ◡(𝑆‘𝐹)) = (𝑆‘(𝐹 ∘ ◡𝐹)))
23		simp3 1063	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑆 ∈ 𝐸 ∧ 𝐹 ∈ 𝑇) → 𝐹 ∈ 𝑇)
24	2, 3	ltrncnv 35432	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) → ◡𝐹 ∈ 𝑇)
25	24	3adant2 1080	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑆 ∈ 𝐸 ∧ 𝐹 ∈ 𝑇) → ◡𝐹 ∈ 𝑇)
26	2, 3, 4	tendospdi1 36309	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑆 ∈ 𝐸 ∧ 𝐹 ∈ 𝑇 ∧ ◡𝐹 ∈ 𝑇)) → (𝑆‘(𝐹 ∘ ◡𝐹)) = ((𝑆‘𝐹) ∘ (𝑆‘◡𝐹)))
27	1, 12, 23, 25, 26	syl13anc 1328	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑆 ∈ 𝐸 ∧ 𝐹 ∈ 𝑇) → (𝑆‘(𝐹 ∘ ◡𝐹)) = ((𝑆‘𝐹) ∘ (𝑆‘◡𝐹)))
28	22, 27	eqtrd 2656	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑆 ∈ 𝐸 ∧ 𝐹 ∈ 𝑇) → ((𝑆‘𝐹) ∘ ◡(𝑆‘𝐹)) = ((𝑆‘𝐹) ∘ (𝑆‘◡𝐹)))
29	28	coeq2d 5284	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑆 ∈ 𝐸 ∧ 𝐹 ∈ 𝑇) → (◡(𝑆‘𝐹) ∘ ((𝑆‘𝐹) ∘ ◡(𝑆‘𝐹))) = (◡(𝑆‘𝐹) ∘ ((𝑆‘𝐹) ∘ (𝑆‘◡𝐹))))
30		coass 5654	. . . 4 ⊢ ((◡(𝑆‘𝐹) ∘ (𝑆‘𝐹)) ∘ ◡(𝑆‘𝐹)) = (◡(𝑆‘𝐹) ∘ ((𝑆‘𝐹) ∘ ◡(𝑆‘𝐹)))
31		coass 5654	. . . 4 ⊢ ((◡(𝑆‘𝐹) ∘ (𝑆‘𝐹)) ∘ (𝑆‘◡𝐹)) = (◡(𝑆‘𝐹) ∘ ((𝑆‘𝐹) ∘ (𝑆‘◡𝐹)))
32	29, 30, 31	3eqtr4g 2681	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑆 ∈ 𝐸 ∧ 𝐹 ∈ 𝑇) → ((◡(𝑆‘𝐹) ∘ (𝑆‘𝐹)) ∘ ◡(𝑆‘𝐹)) = ((◡(𝑆‘𝐹) ∘ (𝑆‘𝐹)) ∘ (𝑆‘◡𝐹)))
33	10	coeq1d 5283	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑆 ∈ 𝐸 ∧ 𝐹 ∈ 𝑇) → ((◡(𝑆‘𝐹) ∘ (𝑆‘𝐹)) ∘ (𝑆‘◡𝐹)) = (( I ↾ (Base‘𝐾)) ∘ (𝑆‘◡𝐹)))
34	2, 3, 4	tendocl 36055	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑆 ∈ 𝐸 ∧ ◡𝐹 ∈ 𝑇) → (𝑆‘◡𝐹) ∈ 𝑇)
35	25, 34	syld3an3 1371	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑆 ∈ 𝐸 ∧ 𝐹 ∈ 𝑇) → (𝑆‘◡𝐹) ∈ 𝑇)
36	6, 2, 3	ltrn1o 35410	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑆‘◡𝐹) ∈ 𝑇) → (𝑆‘◡𝐹):(Base‘𝐾)–1-1-onto→(Base‘𝐾))
37	1, 35, 36	syl2anc 693	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑆 ∈ 𝐸 ∧ 𝐹 ∈ 𝑇) → (𝑆‘◡𝐹):(Base‘𝐾)–1-1-onto→(Base‘𝐾))
38		f1of 6137	. . . 4 ⊢ ((𝑆‘◡𝐹):(Base‘𝐾)–1-1-onto→(Base‘𝐾) → (𝑆‘◡𝐹):(Base‘𝐾)⟶(Base‘𝐾))
39		fcoi2 6079	. . . 4 ⊢ ((𝑆‘◡𝐹):(Base‘𝐾)⟶(Base‘𝐾) → (( I ↾ (Base‘𝐾)) ∘ (𝑆‘◡𝐹)) = (𝑆‘◡𝐹))
40	37, 38, 39	3syl 18	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑆 ∈ 𝐸 ∧ 𝐹 ∈ 𝑇) → (( I ↾ (Base‘𝐾)) ∘ (𝑆‘◡𝐹)) = (𝑆‘◡𝐹))
41	32, 33, 40	3eqtrd 2660	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑆 ∈ 𝐸 ∧ 𝐹 ∈ 𝑇) → ((◡(𝑆‘𝐹) ∘ (𝑆‘𝐹)) ∘ ◡(𝑆‘𝐹)) = (𝑆‘◡𝐹))
42	2, 3	ltrncnv 35432	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑆‘𝐹) ∈ 𝑇) → ◡(𝑆‘𝐹) ∈ 𝑇)
43	1, 5, 42	syl2anc 693	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑆 ∈ 𝐸 ∧ 𝐹 ∈ 𝑇) → ◡(𝑆‘𝐹) ∈ 𝑇)
44	6, 2, 3	ltrn1o 35410	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ◡(𝑆‘𝐹) ∈ 𝑇) → ◡(𝑆‘𝐹):(Base‘𝐾)–1-1-onto→(Base‘𝐾))
45	1, 43, 44	syl2anc 693	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑆 ∈ 𝐸 ∧ 𝐹 ∈ 𝑇) → ◡(𝑆‘𝐹):(Base‘𝐾)–1-1-onto→(Base‘𝐾))
46		f1of 6137	. . 3 ⊢ (◡(𝑆‘𝐹):(Base‘𝐾)–1-1-onto→(Base‘𝐾) → ◡(𝑆‘𝐹):(Base‘𝐾)⟶(Base‘𝐾))
47		fcoi2 6079	. . 3 ⊢ (◡(𝑆‘𝐹):(Base‘𝐾)⟶(Base‘𝐾) → (( I ↾ (Base‘𝐾)) ∘ ◡(𝑆‘𝐹)) = ◡(𝑆‘𝐹))
48	45, 46, 47	3syl 18	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑆 ∈ 𝐸 ∧ 𝐹 ∈ 𝑇) → (( I ↾ (Base‘𝐾)) ∘ ◡(𝑆‘𝐹)) = ◡(𝑆‘𝐹))
49	11, 41, 48	3eqtr3rd 2665	1 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑆 ∈ 𝐸 ∧ 𝐹 ∈ 𝑇) → ◡(𝑆‘𝐹) = (𝑆‘◡𝐹))

Syntax hints:   → wi 4   ∧ wa 384   ∧ w3a 1037   = wceq 1483   ∈ wcel 1990   I cid 5023  ^◡ccnv 5113   ↾ cres 5116   ∘ ccom 5118  ⟶wf 5884  –1-1-onto→wf1o 5887  ‘cfv 5888  Basecbs 15857  HLchlt 34637  LHypclh 35270  LTrncltrn 35387  TEndoctendo 36040

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-map 7859  df-preset 16928  df-poset 16946  df-plt 16958  df-lub 16974  df-glb 16975  df-join 16976  df-meet 16977  df-p0 17039  df-p1 17040  df-lat 17046  df-clat 17108  df-oposet 34463  df-ol 34465  df-oml 34466  df-covers 34553  df-ats 34554  df-atl 34585  df-cvlat 34609  df-hlat 34638  df-lhyp 35274  df-laut 35275  df-ldil 35390  df-ltrn 35391  df-trl 35446  df-tendo 36043

This theorem is referenced by:  tendospcanN  36312  dihjatcclem4  36710